Descibe the sample space :
A coin is tossed . If it results in a head , a die is thrown . If the die is shown up an even number the die is thrown again.

Answer

To describe the sample space for the given scenario, we will break down the problem step by step.

### Step 1: Toss the Coin
The first step is to toss a coin. The possible outcomes from tossing a coin are:
- Head (H)
- Tail (T)

### Step 2: If the Coin Shows Head
If the outcome of the coin toss is a Head (H), we proceed to the next step, which is to roll a die. The possible outcomes when rolling a die are:
- 1, 2, 3, 4, 5, 6

### Step 3: Determine Even or Odd Outcome from the Die
When the die is rolled, we can categorize the outcomes into even and odd:
- Even numbers: 2, 4, 6
- Odd numbers: 1, 3, 5

### Step 4: If the Die Shows an Even Number
If the die shows an even number (2, 4, or 6), we roll the die again. The outcomes for this second roll are again:
- 1, 2, 3, 4, 5, 6

### Step 5: Constructing the Sample Space
Now, we can construct the sample space based on the outcomes:

1. If the coin shows T (Tail), the outcome is simply T.
2. If the coin shows H (Head), we have the following scenarios based on the first die roll:
- If the first die roll is 1 (odd): Outcomes are H1
- If the first die roll is 2 (even): Outcomes are H2 followed by a second die roll (1, 2, 3, 4, 5, 6), giving us:
- H2,1
- H2,2
- H2,3
- H2,4
- H2,5
- H2,6
- If the first die roll is 3 (odd): Outcomes are H3
- If the first die roll is 4 (even): Outcomes are H4 followed by a second die roll (1, 2, 3, 4, 5, 6), giving us:
- H4,1
- H4,2
- H4,3
- H4,4
- H4,5
- H4,6
- If the first die roll is 5 (odd): Outcomes are H5
- If the first die roll is 6 (even): Outcomes are H6 followed by a second die roll (1, 2, 3, 4, 5, 6), giving us:
- H6,1
- H6,2
- H6,3
- H6,4
- H6,5
- H6,6

### Final Sample Space
Combining all the outcomes, the complete sample space is:
- S = {T, H1, H2, H2,1, H2,2, H2,3, H2,4, H2,5, H2,6, H3, H4, H4,1, H4,2, H4,3, H4,4, H4,5, H4,6, H5, H6, H6,1, H6,2, H6,3, H6,4, H6,5, H6,6}

### Summary of the Sample Space
The sample space consists of:
- One outcome for Tail (T)
- One outcome each for Heads with odd results (H1, H3, H5)
- Multiple outcomes for Heads with even results followed by second rolls (H2, H4, H6 with their respective second roll outcomes)